Problems and Solutions on Thermodynamics and Statistical Mechanics

168 Problems €4 Solutiona on Therrncdynamica d Statistical Mechanics

If E/& >> 1, N - E/& >> 1, we have

N

r 1

=k ICE\. -In-+ "E" ( 1-- :E) ln- 'El'

1--

EN

2009

Consider a system of N non-interacting particles, each fixed in position

and carrying a magnetic moment p, which is immersed in a magnetic field

H. Each particle may then exist in one of the two energy states E = 0 or

E = 2pH. Treat the particles as distinguishable.

(a) The entropy, S, of the system can be written in the form S =

klnR(E), where k is the Boltzmann constant and E is the total system

energy. Explain the meaning of R(E).

(b) Write a formula for S(n), where n is the number of particles in the

upper state. Crudely sketch S(n).

(c) Derive Stirling's approximation for large n:

Inn! = nlnn - n

by approximating In n! by an integral.

(d) Rewrite the result of (b) using the result of (c). Find the value of

(e) Treating E as continuous, show that this system can have negative

(f) Why is negative temperature possible here but not for a gas in a

(CUSPEA)

(a) R(E) is the number of all the possible microscopic states of the

n for which S(n) is maximum.

absolute temperature.

box?

Solution:

system when its energy is E, where